A superspace formalism for general relativity was initially introduced for the purposes of quantizing gravity and in particular quantum cosmology.
It originates from canonical quantization of gravity by means of functional Wheeler-De Witt Schrödinger type equation [1]}. For Bianchi cosmological models however it is possible to simplify some considerations by replacing
infinite dimensional (functional) superspace by a finite dimensional minisuperspace. Thus MSS approach reduced the quantum cosmology to quantum mechanical systems of single particle with different potentials and kinetic energy terms as a toy model see e.g. [2]}-[3]}.
Later on the corresponding classical mechanical system in MSS formulation
proved to be extremely effective in studying (classical and quantum) integrability of scalar-tensor cosmological models
in a specific (usually Einstein or Jordan) frame [4]}-[5]}, and finding out their exact solutions,
see e.g. [6]}-[7]} and references therein. Hamiltonian framework is also very useful in dynamical system analysis of such systems.
Our aim in the present paper is to extend these considerations to a broader frame-independent context.
